Nuclear Magnetic Resonance (NMR) tools used for well-logging or downhole fluid characterization measure the response of nuclear spins in formation fluids to applied magnetic fields. Downhole NMR tools typically have a permanent magnet that produces a static magnetic field at a desired test location (e.g., where the fluid is located). The static magnetic field produces a non-equilibrium magnetization in the fluid. The magnetization is aligned along the direction of the static field. The magnitude of the induced magnetization is proportional to the magnitude of the static field. The proportionality constant is the static magnetic susceptibility. A transmitter antenna produces a time-dependent radio frequency magnetic field that is perpendicular to the direction of the static field. The NMR resonance condition is satisfied when the radio frequency is equal to the Larmor frequency, which is proportional to the magnitude of the static magnetic field. The radio frequency magnetic field produces a torque on the magnetization vector that causes it to rotate about the axis of the applied radio frequency field. The rotation results in the magnetization vector developing a component perpendicular to the direction of the static magnetic field. This causes the magnetization vector to precess around the static field at the Larmor frequency. At resonance between the Larmor and transmitter frequencies, the magnetization is tipped to the transverse plane (i.e., a plane normal to static magnetic field vector). A series of radio frequency pulses are applied to generate spin echoes that are measured with the antenna.
Crude oil properties such as viscosity, molecular composition, gas-oil ratio, and SARA fractions are crucial parameters for evaluating, for example, reservoir quality, producibility, and compartmentalization. In the past decade, physical and empirical model-based equations have been developed which relate the properties of crude oils to Nuclear Magnetic Resonance (NMR) measurements. However, in general, the existing models are too simplistic to accurately estimate crude oil properties. This limitation arises because of the inherent complexity of crude oils. That is, they are mixtures of hydrocarbon and non-hydrocarbon molecules of varying shapes, sizes, and concentrations. There are also shortcomings in other database approaches such as, for example, Artificial Neural Networks (ANN). Implementation of ANN requires computationally expensive and lengthy iterative training that may not converge to a solution.
Characterization of reservoir fluids is crucial for several aspects of reservoir development and management. For example, fluid properties such as viscosity and molecular composition are used to calculate flow rates and sweep efficiencies of secondary and tertiary recoveries. Gas-oil ratio (GOR) of reservoir fluids is an important parameter for material selection of well completion and design of surface facilities. Asphaltene and wax concentrations are key considerations for flow assurance in completions, pipelines, and surface facilities. Estimation of fluid properties at different depths in a reservoir provides indications of compositional grading and compartmentalization within the reservoir. The direct measurement of fluid properties in a laboratory, however, is time consuming and expensive. As a result, it is useful to estimate fluid properties from measurements such as NMR which can be performed with relative ease and at downhole temperature and pressure conditions.
NMR response of fluids provides a link between microscopic molecular motions and macroscopic properties such as viscosity and composition. The relationship between viscosity and relaxation time of pure fluids was established by the phenomenological relaxation theory of Bloembergen, Purcell, and Pound (BPP). Brown studied proton relaxation in a suite of crude oils with various compositions and viscosities. The viscosities of the samples varied from about 0.5 to 400 cp. He found that the relaxation times showed an inverse dependence on viscosity over the entire range. Since the early work of Brown, several physical and empirical models have been proposed that relate crude oil properties to NMR response. However, the predictive power of these models is limited for several reasons. First, crude oils are complex mixtures of linear, branched, cyclic, and aromatic hydrocarbons. They also contain compounds with sulfur, oxygen, and nitrogen atoms, in addition to small concentrations of metallic impurities such as nickel and vanadium. As a result, the NMR response of crude oils is governed by a multitude of intra- and inter-molecular interactions between the constituents. It is difficult to accurately describe all such interactions by simple physical or empirical models. Second, the detailed information contained in the shapes of T1 or T2 distributions is not accounted for in the models. Last, the empirical constants involved in the models are not universal, and may differ by as much as a factor of two for different oils.
Radial basis functions (RBFs) are used for several applications in numerical and scientific computing, such as solution of partial differential equations, artificial neural networks, surface reconstruction, computer-aided-design, computer graphics, and multivariate interpolation. A unique property of RBFs is that they provide excellent interpolants for high dimensionality data sets of poorly distributed data points. This property follows from the mathematical result that a linear system of interpolation equations with RBFs is invertible under very mild conditions. The theoretical background for the invertibility of the RBF interpolation matrix has been established and a study evaluating 29 interpolating methods concluded that interpolation by multiquadric RBFs outperformed most methods. The application of RBFs for numerical solution of elliptic, hyperbolic, and parabolic differential equations has been developed and RBFs have been extensively used to approximate scattered, non-uniformly distributed data.
The traditional approach to solve an inverse problem involves fitting a theoretical or empirical model to the measurements. This approach is not generally suited to petrophysical systems because they are too complex to be parameterized by forward models. For example, the complex molecular interactions that govern NMR relaxation of crude oils can not be fully described by simple forward models. A technique using RBFs to solve complex inverse problems for which accurate forward models are unknown has been previously introduced, demonstrated, and used, inter alia, to predict viscosities and molecular compositions of dead oils from NMR measurements.